Principle of Relativity and The Renormalizable Quantum Gravity

TL;DR

The paper develops a quantum principle of relativity (QPR) that replaces diffeomorphism invariance and yields a unified Hilbert space with a quantum-relativity (QR) constraint, $\hat{P}\,|\Psi\rangle_{\mathrm{QR}} = 0$, ensuring consistent active-passive transformations across quantum reference frames. By promoting the metric to a quantum field within a path-integral framework and introducing a ghost-like graviton $\lambda_{\alpha\beta}$, the authors derive graviton propagators as causal, internal degrees of freedom and demonstrate a potentially renormalizable quantum gravity that reduces to general relativity at large distances. Renormalization is addressed via a scale hierarchy involving a curvature-precision parameter $\phi_G$ and a Lorentz-invariant cutoff, with explicit calculations of gravitational self-energy and finite-temperature behavior for a scalar as validation examples. The framework offers a novel route to quantum gravity that preserves relational observables, avoids traditional diffeomorphism-based issues, and provides concrete, testable predictions such as mass-running effects and Hawking-like thermodynamics within a renormalizable QR setting.

Abstract

We develop a purely quantum theory based on the novel principle of relativity, termed the quantum principle of relativity, instead of applying the diffeomorphism invariance. We demonstrate that the essence of the principle can be extended into the quantum realm, maintaining the identical structures of active and passive transformations. By employing this principle, we show that quantum gravitational effects are naturally realized within the renormalizable theory, with general relativity emerging in large distances. We derive graviton propagators and provide several examples grounded in this novel framework.

Figures (5)

• Figure 1: Graphical representation of Eq. \ref{['eq:rulerA']} (above) and Eq. \ref{['eq:rulerC']} (below), assuming $x=z_1=0$ for simplicity. Ruler B is not included, as it is not involved in the reference switching process. Both cases represent the same physical situation; only the coordinate values differ, while the relative distances are preserved.
• Figure 2: Diagram of the quantum experiment in spacetime. The quantum system is assumed to be well-regulated within the experimental setup, ensuring a finite volume despite its vast size relative to the quantum scale. The system's spatial boundary, where the measurement devices are located, is constantly monitored to observe particles escaping from it. The experiment begins by identifying the initial conditions of the fields, and ends with the measurement of the fields across the space.
• Figure 3: Example diagrams for group A (above) and group B (below), respectively. The left figures show the diagrams when the graviton channels are turned off, while the right figures show the complete diagrams. Each interaction vertex has its coupling constant, $g_n$, which can be either kinetic ($\propto \kappa p_{n\alpha} p_{n\beta}$) or massive ($\propto \kappa m^2$).
• Figure 4: Graviton-involved 2-point loops in QR free scalar theory. (a) Two possible 1PI loops, including graviton-graviton-scalar loop and graviton-scalar-scalar loop. Both loops have a diverging order of $O(\kappa^2 \mu^6)$. (b) Particle oscillation between gravitons and scalar particles. As the number of oscillation periods increases, the highest diverging order of a single oscillation unit also increases and eventually converges to $O(\kappa^2 \mu^6)$. (c) Graviton-scalar loops fully formed in a scalar vacuum bubble. In such cases, the highest diverging order is $O((\kappa \mu^3)^n)$, when the size of the vacuum bubble is denoted by $n$.
• Figure 5: Infinite summation of graviton-graviton-scalar loops. The index $n$ denotes a number of involved loops. The interaction couplings, $g_n$, can be either massive or kinetic, resulting in each loop having four possible combinations: massive-massive, kinetic-kinetic, kinetic-massive, and massive-kinetic.